[Paper Review] Affine-Invariant Integrated Rank-Weighted Depth: Definition, Properties and Finite Sample Analysis

Guillaume Staerman, Pavlo Mozharovskyi+1 more|arXiv (Cornell University)|Jun 21, 2021

Advanced Statistical Methods and Models4 citations

TL;DR

This paper proposes Affine-Invariant Integrated Rank-Weighted Depth (AI-IRW), a novel multivariate depth function that extends the IRW depth by incorporating the precision matrix to ensure affine-invariance, thereby satisfying all four key axioms of statistical depth. The method achieves robustness and computational efficiency, with theoretical concentration bounds and strong empirical performance in anomaly detection and rank coherence under contamination.

ABSTRACT

Because it determines a center-outward ordering of observations in $\\mathbb{R}^d$ with $d\\geq 2$, the concept of statistical depth permits to define quantiles and ranks for multivariate data and use them for various statistical tasks (e.g. inference, hypothesis testing). Whereas many depth functions have been proposed \ extit{ad-hoc} in the literature since the seminal contribution of \\cite{Tukey75}, not all of them possess the properties desirable to emulate the notion of quantile function for univariate probability distributions. In this paper, we propose an extension of the \ extit{integrated rank-weighted} statistical depth (IRW depth in abbreviated form) originally introduced in \\cite{IRW}, modified in order to satisfy the property of \ extit{affine-invariance}, fulfilling thus all the four key axioms listed in the nomenclature elaborated by \\cite{ZuoS00a}. The variant we propose, referred to as the Affine-Invariant IRW depth (AI-IRW in short), involves the covariance/precision matrices of the (supposedly square integrable) $d$-dimensional random vector $X$ under study, in order to take into account the directions along which $X$ is most variable to assign a depth value to any point $x\\in \\mathbb{R}^d$. The accuracy of the sampling version of the AI-IRW depth is investigated from a nonasymptotic perspective. Namely, a concentration result for the statistical counterpart of the AI-IRW depth is proved. Beyond the theoretical analysis carried out, applications to anomaly detection are considered and numerical results are displayed, providing strong empirical evidence of the relevance of the depth function we propose here.

Motivation & Objective

• To address the lack of affine-invariance in the original IRW depth, which makes depth values sensitive to coordinate system choices.
• To develop a depth function that satisfies all four axioms of statistical depth, including affine-invariance, as defined by Zuo & Serfling (2000).
• To ensure the depth function remains robust and stable under contamination and sampling variability, particularly in high-dimensional settings.
• To provide a non-asymptotic finite-sample analysis of the sampling version of the AI-IRW depth, including concentration bounds.
• To demonstrate the method’s practical utility in anomaly detection and rank ordering under various distributional and contamination scenarios.

Proposed method

• The AI-IRW depth is defined as the IRW depth of the transformed random vector $\Sigma^{-1/2}X$, where $\Sigma$ is the covariance matrix of $X$, ensuring affine-invariance through standardization via the precision matrix.
• The method uses Monte Carlo approximation over uniformly sampled directions on the unit sphere to compute the depth, enabling efficient empirical estimation.
• The statistical counterpart of AI-IRW is analyzed via non-asymptotic concentration inequalities, establishing finite-sample reliability.
• Robust covariance estimation via MCD (Minimum Covariance Determinant) and SC (Sample Covariance) is integrated to enhance stability under contamination.
• The depth is evaluated using Kendall’s $\tau$ distance to measure rank coherence between clean and corrupted data sets.
• The method is applied to anomaly detection and compared against IRW, halfspace depth, and halfspace mass depth on Gaussian and heavy-tailed (Student-3) distributions.

Experimental results

Research questions

• RQ1 Does the proposed AI-IRW depth satisfy the four key axioms of statistical depth, particularly affine-invariance, which the original IRW depth fails to meet?
• RQ2 How does the finite-sample version of AI-IRW perform in terms of concentration and stability under sampling variability?
• RQ3 To what extent does AI-IRW maintain rank coherence and robustness when data are contaminated with outliers?
• RQ4 How does the use of robust covariance estimators (MCD vs. SC) affect the stability and performance of AI-IRW?
• RQ5 How does AI-IRW compare empirically to existing depth functions in anomaly detection and rank ordering tasks?

Key findings

• The AI-IRW depth satisfies all four axioms of statistical depth, including affine-invariance, by transforming the data using the precision matrix $\Sigma^{-1/2}$.
• A non-asymptotic concentration inequality is established for the sampling version of AI-IRW, ensuring reliable finite-sample performance.
• Under contamination, AI-IRW with MCD estimator maintains high rank coherence (Kendall $\tau$), while IRW and SC-based AI-IRW degrade rapidly after 1% outliers.
• The AI-IRW depth shows comparable or lower variance than halfspace and halfspace mass depth across multiple sample realizations and noisy direction settings.
• The method achieves strong empirical performance in anomaly detection, with high coherence in rank ordering even under heavy-tailed and contaminated distributions.
• The use of MCD estimator provides robustness to AI-IRW, but the underlying robustness of IRW limits further improvement, indicating a 'worst-case' robustness trade-off.